How do I turn an imported SVG into a sphere?
There is no quick 'one-button-press' way to do what you want easily, especially not directly from the SVG imported Curve Objects. Curve objects are not very suited for modeling other than 'extrusion based shapes.

You can however quickly simulate an approximation of a 3D object by projection. It will yield very bad topology, and the resulting object will not be very suited for further modelling, but visually it may satisfy your needs.

Import your SVG curves, fix their center with the 3D cursor, and center them at the scene origin.



Add two sphere objects and scale one to fit the largest radius, the other to fit the smallest radius. Move the logos above your spheres





By this specific order select the matching curve > the select the corresponding sphere radius.
Enter Edit Mode, select all faces press the Spacebar Menu > search for Knife Project and then tick the option Cut Through.



Press Ctrl + I to invert selection and quickly remove the unwanted faces.



Repeat with the other curves for the corresponding spheres.



Assign materials to chunks of your mesh. Press L while hovering a part of the mesh to select all linked geometry



In the end add a Solidify modifier and an EdgeSplit modifier to your objects.



Have in mind that this is an approximation, and will not necessarily look good viewed from all angles.